Abstract
We consider the Erdős–Rényi random graph G(n, p) inside the critical window, that is when p = 1/n + λn −4/3, for some fixed \({\lambda \in \mathbb{R}}\) . We prove that the sequence of connected components of G(n, p), considered as metric spaces using the graph distance rescaled by n −1/3, converges towards a sequence of continuous compact metric spaces. The result relies on a bijection between graphs and certain marked random walks, and the theory of continuum random trees. Our result gives access to the answers to a great many questions about distances in critical random graphs. In particular, we deduce that the diameter of G(n, p) rescaled by n −1/3 converges in distribution to an absolutely continuous random variable with finite mean.
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L. Addario-Berry was supported by an NSERC Discovery Grant throughout the research and writing of this paper. C. Goldschmidt was funded by EPSRC Postdoctoral Fellowship EP/D065755/1.
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Addario-Berry, L., Broutin, N. & Goldschmidt, C. The continuum limit of critical random graphs. Probab. Theory Relat. Fields 152, 367–406 (2012). https://doi.org/10.1007/s00440-010-0325-4
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DOI: https://doi.org/10.1007/s00440-010-0325-4